A molecule has a variety of vibration modes. When radiation of the same frequency as one of these modes hits the molecule, the molecule vibrates and then reradiates that energy but in a random direction. Thus some of the impinging radiation will be radiated back in the direction it came from. This is the mechanism of the greenhouse effect. Radiation which is not of a frequency equal to one of the vibration modes of the molecule does not interact with the molecules.

The Broadening of the Spectral Lines

The discrete set of vibration frequencies of a molecule is called its spectrum; this is both a la Kirchhoff's Law its absorption spectrum and its emissions spectrum. If the impinging radiation had to have exactly the wavelength of the discrete spectral lines there would not be much interaction between the radiation an the molecules.

The spectrum is modified by the motion of the molecules. The Doppler effect is the modification of the perceived frequency of radiation due to the motion of the molecule. If the molecule is traveling in opposite direction from the incoming radiation the perceived frequency of the radiation is greater. Thus if radiation were slightly lower frequency than a vibration frequency of a molecule the Doppler effect could bring about a coincidence with the vibration frequency of the molecule. If a molecule were traveling in the same direction as incoming radiation the Doppler effect lowers its perceived frequency and thus could result in the absorption of radiation of a slightly higher frequency.

In effect the lines of the absorption spectrum are broadened by the Doppler effect. They are also broadened by collision frequency within a gas.

The character of radiation can be expressed by its frequency ω or its wavelength λ. These two measures are inversely related, their product being equal to velocity c of the radiation; i.e.,

ωλ = c
 

The frequency of radiation is expressed in cycles per second but a more convenient representation for spectroscopic analysis is the reciprocal of the wavelength, a quantity called the wave number of the radiation. It is usually expressed in inverse centimeters, cm^-1. Let ν represent the frequency of radiation expressed as wave number and ν₀ be the wave number of an absorption line for a molecule. The absorption coefficient κ(ν) takes the form called the Lorentz line shape

κ(ν) = Sγ/[π((ν-ν₀)²+γ²]
 

where S is called the line strength and γ is a parameter called the half-width of the line. This line shape is shown below:

The half-width of the line is given by

γ = 1/(2πτc)
 

where c is the velocity of the radiation and τ is the average time between collisions for a molecule in the gas. This average time between collisions is inversely proportion to the pressure of the gas. Thus

γ = γ₀p/p₀
 

where p is pressure and p₀ is standard pressure (1000 mb). The value of γ₀ is approximately 0.1 cm^-1.

When there are two spectral lines the absorption coefficient function is as shown below:

The amount of radiation absorbed depends upon the spectral distribution of the impinging radiation. For example, consider the infrared radiation eminating for surfaces of three different temperature. Suppose the gas absorbs radiation around wavelenth λ. As illustrated below the amount of energy in the three radiation coming from the three surfaces in the vicinity of wavelength λ is different. In these diagrams the frequency increases from left to right, but the wavelength decreases from left to right.

In this case the proportion of radiation absorbed goes up as the temperature goes down.

What the above means is that the radiative efficiency of a gas is not an intrinsic characteristic of the substance. It does depend upon the line spectrum of the substance but the broadening of that spectrum is a function of pressure and temperature. And, of course, the amount of radiation absorbed depends upon the spectrum of the radiation impinging upon the gas.

The dependence of the absorption depending upon pressure and temperature is in the nature of a positive feedback. On Mars the atmosphere is carbon dioxide but at a low pressure and temperature. Therefore the absorption spectrum of carbon dioxide there is not much broadened and the greenhouse effect is even less than what is accounted for by the low density. On the other hand, on Venus the high pressure and temperature broadens the absorption spectrum of carbon dioxide so it is a more effective greenhouse gas.

(To be continued.)